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I want to find limit of $$ \int_{0}^{\infty}n{\rm e}^{-nx} \sin\left(\frac{1}{x}\right) {\rm d}x\quad\mbox{as}\quad n\to\infty $$ if it exists or to prove that it doesn't exist.

  • I see that $n{\rm e}^{-nx}\sin\left(1/x\right)\to 0,\ \forall\ x > 0$ and that the convergence is uniform on $\left[a,\infty\right),\ \forall\ a > 0$.
    • That implies $$ \int_{a}^{\infty}n{\rm e}^{-nx}\sin\left(\frac{1}{x}\right){\rm d}x \to 0\quad\mbox{as}\quad n\to\infty,\ \forall\ a > 0 $$

Can anyone tell me what the next step is or if I'm on the wrong track ?. Thanks.

Felix Marin
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Fry
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    As you are tagging this "measure-theory", why don't you consider using dominated convergence theorem? Uniform convergence may not be a good idea since swapping integration/limit may not hold for unbounded interval, even under uniform convergence. –  Aug 24 '11 at 21:48
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    @user, if you recommend to use that there exists a so-called specific integer $m$ such that $ne^{-nx}\le me^{-mx}$ uniformly over $x$ and $n$ (or maybe every $n$ large enough), then you might reconsider: this is false. – Did Aug 24 '11 at 22:23
  • @Didier: Oh yeah, you are right. Sorry about that. – user1736 Aug 25 '11 at 00:36

1 Answers1

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Change variables $y = nx$. Your integral becomes $$\int_0^{\infty}e^{-y}\sin({n \over y})\,dy$$ $$= {{1 \over n}}\int_0^{\infty}y^2e^{-y}{{n \over y^2}}\sin({n \over y})\,dy$$ Integrate this by parts, integrating ${n \over y^2}\sin({n \over y})$ to $\cos({n \over y})$, and differentiating $y^2e^{-y}$. One obtains $${1 \over n}\lim_{y \rightarrow \infty} \big(y^2 e^{-y}\cos({n \over y})\big) - {1 \over n}\lim_{y \rightarrow 0} \big(y^2 e^{-y}\cos({n \over y})\big) - {1 \over n}\int_0^{\infty}(2y - y^2)e^{-y}\cos({n \over y})\,dy$$ Observing that both the boundary terms go to zero, this becomes $$- {1 \over n}\int_0^{\infty}(2y - y^2)e^{-y}\cos({n \over y})\,dy$$ Since $|\cos({n \over y})| \leq 1$ for all $n$ and all $y$, the above is bounded in absolute value by $${1 \over n}\int_0^{\infty}(2y + y^2)e^{-y}\,dy$$ Since the integral doesn't depend on $n$ and is finite, the limit as $n$ goes to infinity of the above expression is zero.

Zarrax
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